$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ). $2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them

Maria has a $6 \times 5$ board with some shaded squares, as in the figure. She writes, in some order, the digits $1, 2, 3, 4$ and $5$ in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row. For example, if you wrote $2, 3, 4, 1, 5$ in the first row, then since $4$ is in the shaded box, the number that occupies the fourth place $(1)$ is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: $2, 3, 4, 5, 1$. Then, to complete the third row, as in the shaded box is $3$, the number located in the third place $(4)$ writes it in the last box and gets $2, 3, 5, 1, 4$. Following in the same way, he gets the board of the figure. Show a way to locate the numbers in the first row to get the numbers $2, 4, 5, 1, 3$ in the last row.

At the start of the game there are $ r$ red and $ g$ green pieces/stones on the table. Hojoo and Kestutis make moves in turn. Hojoo starts. The person due to make a move, chooses a colour and removes $ k$ pieces of this colour. The number $ k$ has to be a divisor of the current number of stones of the other colour. The person removing the last piece wins. Who can force the victory?

[b]p1.[/b] Find all real solutions of the system $$x^5 + y^5 = 1$$ $$x^6 + y^6 = 1$$ [b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the circles, which are outside of the triangle. [b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits. [b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves? [b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer. [b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer. PS. You should use hide for answers.

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$, the $2n-1$ cells on $i$. row and $i$. column have all different colors. a) Prove that for $n=2001$ and $k=4001$, such coloring is not possible. b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$, such coloring is possible.

Consider a finite collection of 3-element sets $A_i$, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each $A_i$ has at least one red element.

There is a black token in the lower-left corner of a board $m \times n$ ($m, n \ge 3$), and there are white tokens in the lower-right and upper-left corners of this board. Petryk and Vasyl are playing a game, with Petryk playing with a black token and Vasyl with white tokens. Petryk moves first. In his move, a player can perform the following operation at most two times: choose any his token and move it to any adjacent by side cell, with one restriction: you can't move a token to a cell where at some point was one of the opponents' tokens. Vasyl wins if at some point of the game white tokens are in the same cell. For which values of $m, n$ can Petryk prevent him from winning? [i](Proposed by Arsenii Nikolaiev)[/i]

$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule: 1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students; 2) Each student received the maximum possible points in each problem or got $ 0$ in it; Lasha got the least number of points. What's the maximal number of points he could have? Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 \minus{} k$ points in it.

[b]p1.[/b] $17.5\%$ of what number is $4.5\%$ of $28000$? [b]p2.[/b] Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$. The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] In the $xy$-plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$. Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at? [b]p4.[/b] What are the last two digits of the sum of the first $2021$ positive integers? [b]p5.[/b] A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] How many terms are in the arithmetic sequence $3$, $11$, $...$, $779$? [b]p7.[/b] Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$? [b]p8.[/b] What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$? [b]p9.[/b] Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$, $AB = 10$, $BC = 9$, and the area of $\vartriangle ABC$ is $36$. Compute the length of $AC$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png[/img] [b]p10.[/b] If $x + y - xy = 4$, and $x$ and $y$ are integers, compute the sum of all possible values of$ x + y$. [b]p11.[/b] What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap? [b]p12.[/b] $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$. Compute the smallest possible value of $N$. [b]p13.[/b] Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years? [b]p14.[/b] Say there is $1$ rabbit on day $1$. After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$? [b]15.[/b] Ajit draws a picture of a regular $63$-sided polygon, a regular $91$-sided polygon, and a regular $105$-sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have? [b]p16.[/b] Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p17.[/b] Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$'s adjacent. [b]p18.[/b] From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$, where $a$ and $b$ are positive integers. Compute $a + b$. [b]p19.[/b] Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$. He starts by putting the first marble in bucket $1$, the second marble in bucket $2$, the third marble in bucket $3$, etc. After placing a marble in bucket $9$, he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag? [img]https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png[/img] [b]p20.[/b] What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a $100 \times 100$ square grid $G$. Initially color all cells of $G$ white. A move consists of selecting a $1 \times 7$ or $7 \times 1$ subgrid of $G$ and flipping the colors of all cells in this subgrid from white to black or vice versa. Is it possible that after a series of moves, all cells are colored black? [i]Proposed by Evan Chang (squareman), USA[/i]

Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$. Prove that there is a regular $n$-gon whose all vertices lie on the considered arcs

Suppose $P$ is a point inside a convex $2n$-gon, such that $P$ does not lie on any diagonal. Show that $P$ lies inside an even number of triangles whose vertices are vertices of the polygon.

Timi was born in $1999$. Ever since her birth how many times has it happened that you could write that day’s date using only the digits $0$, $1$ and $2$? For example, $2022.02.21$. is such a date.

Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

Given is a convex polygon with $n\geq 5$ sides. Prove that there exist at most $\displaystyle \frac{n(2n-5)}3$ triangles of area 1 with the vertices among the vertices of the polygon.

A square of side $n \ge 2$ is divided into $n^2$ unit squares ($n \in N$). One draws $n-1$ lines so that the interior of each of the unit squares is cut by at least one of these lines. (a) Give an example of such a configuration for some $n$. (b) Show that some two of the lines must meet inside the square.

The teacher wrote a non-zero natural number on the board. The teacher explained students that they can delete the number written on the board and can write a number instead naturally new, whenever they want, applying one of the following each time rules: 1) Instead of the current number $n$ write $3n + 13$ 2) Instead of the current number $n$ write the number $\sqrt{n}$, if $n$ is a perfect square . a) If the number $256$ was originally written on the board, is it possible that after a finite number of steps to get the number $55$ on the board? b) If the number $55$ was originally written on the board, is it possible that after a number finished the steps to get the number $256$ on the board?

Nico wants to write the $100$ integers from $1$ to $100$ around a circle in some order and without repetition, such that they have the following property: when moving around the circle clockwise, the sum of the $100$ distances between each number and its next number is equal to $198$. Determine in how many ways the $100$ numbers can be ordered so that Nico achieves his goal. [b]Note:[/b] The distance between two numbers $a$ and $b$ is equal to the absolute value of their difference: $|a - b|$.

In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.

Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$ $n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$ and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$ such that $A_{k+2} = A_{k}.$